تقرير
On the near soliton dynamics for the 2D cubic Zakharov-Kuznetsov equations
| العنوان: | On the near soliton dynamics for the 2D cubic Zakharov-Kuznetsov equations |
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| المؤلفون: | Chen, Gong, Lan, Yang, Yuan, Xu |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Analysis of PDEs |
| الوصف: | In this article, we consider the Cauchy problem for the cubic (mass-critical) Zakharov-Kuznetsov equations in dimension two: $$\partial_t u+\partial_{x_1}(\Delta u+u^3)=0,\quad (t,x)\in [0,\infty)\times \mathbb{R}^{2}.$$ For initial data in $H^1$ close to the soliton with a suitable space-decay property, we fully describe the asymptotic behavior of the corresponding solution. More precisely, for such initial data, we show that only three possible behaviors can occur: 1) The solution leaves a tube near soliton in finite time; 2) the solution blows up in finite time; 3) the solution is global and locally converges to a soliton. In addition, we show that for initial data near a soliton with non-positive energy and above the threshold mass, the corresponding solution will blow up as described in Case 2. Our proof is inspired by the techniques developed for mass-critical generalized Korteweg-de Vries equation (gKdV) equation in a similar context by Martel-Merle-Rapha\"el. More precisely, our proof relies on refined modulation estimates and a modified energy-virial Lyapunov functional. The primary challenge in our problem is the lack of coercivity of the Schr\"odinger operator which appears in the virial-type estimate. To overcome the difficulty, we apply a transform, which was first introduced in Kenig-Martel [13], to perform the virial computations after converting the original problem to the adjoint one. Th coercivity of the Schr\"odinger operator in the adjoint problem has been numerically verified by Farah-Holmer-Roudenko-Yang [9]. Comment: 65 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2407.00300 |
| رقم الأكسشن: | edsarx.2407.00300 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |